1
JEE Main 2024 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1

Let $$a$$ be the sum of all coefficients in the expansion of $$\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$$ and $$b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)$$. If the equation $$c x^2+d x+e=0$$ and $$2 b x^2+a x+4=0$$ have a common root, where $$c, d, e \in \mathbb{R}$$, then $$\mathrm{d}: \mathrm{c}:$$ e equals

A
$$2: 1: 4$$
B
$$1: 1: 4$$
C
$$1: 2: 4$$
D
$$4: 1: 4$$
2
JEE Main 2024 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1

Suppose $$2-p, p, 2-\alpha, \alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1+x)^n$$. Then the value of $$p^2-\alpha^2+6 \alpha+2 p$$ equals

A
8
B
4
C
6
D
10
3
JEE Main 2024 (Online) 27th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if :
A
$2 \sqrt{2}<\mathrm{k}<2 \sqrt{3}$
B
$2 \sqrt{2}<\mathrm{k} \leq 3$
C
$2 \sqrt{3}<\mathrm{k}<3 \sqrt{3}$
D
$2 \sqrt{3}<\mathrm{k} \leq 3 \sqrt{2}$
4
JEE Main 2024 (Online) 27th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :
A
$\mathrm{B}=\mathrm{A}^3$
B
$3 \mathrm{A}=\mathrm{B}$
C
$A=3 B$
D
$\mathrm{A}=\mathrm{B}^3$
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